How to find Analytic Function (Real Part is given in polar form) | Ekeeda.com

Harmonic Function:
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplace's equation.
Examples:
Harmonic functions that arise in physics are determined by their singularities and boundary conditions (such as Dirichlet boundary conditions or Neumann boundary conditions). On regions without boundaries, adding the real or imaginary part of any entire function will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution unique in physical situations by requiring that the solution goes to 0 as you go to infinity. In this case, uniqueness follows by Liouville's theorem.
Analytic function:
An analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. A function is analytic if and only if its Taylor series about x0 converges to the function in some neighborhood for every x0 in its domain.
Examples:
• All polynomials: if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its own Maclaurin series .
• The exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x0 (as in the definition) but for all values of x (real or complex).
• The trigonometric functions, logarithm, and the power functions are analytic on any open set of their domain.

Conjugate function:
A concept in the theory of functions which is a concrete image of some involutory operator for the corresponding class of functions.
1) The function conjugate to a complex-valued function is the function whose values are the complex conjugates of those of .
2) For the function conjugate to a harmonic function see Conjugate harmonic functions.
3) The function conjugate to a -periodic summable function on
the function inverse to . Definition (*) was originally proposed by S. Mandelbrojt for one-dimensional functions, by W. Fenchel in the finite-dimensional case, and by J.

Milne-Thomson method:
The Milne-Thomson method is a method of finding a holomorphic function, whose real or imaginary part is given. The method greatly simplifies the process of finding the holomorphic function whose real or imaginary or any combination of the two parts is given. It is named after Louis Melville Milne-Thomson.

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[XYZ-hihk]

Please sir, my exam is after 6 days. Please please make videos in next 4 questions:

1.Show that: u(r, θ)= e^-θ cos(log r) is harmonic and find its harmonic conjugate, also corresponding analytic function.

2. Show that: u= r^3 cos3θ+rsinθ is harmonic and find its harmonic conjugate.

3.find harmonic conjugate for: v=r^n sin nθ

4.find harmonic conjugate for: v=r^2 sin2θ-1/r sinθ

Please sir 🙏🙏🙏🙏🙏🙏🙏🙏

[maazosmany]

Cant listen wat sir is saying because of music is video it is so irrritating 😤